A Sierpinski number is a positive, odd integer
k for which the integers
k.2n+1
are allcomposite (that is, for every
positive integer n). In 1960
Sierpinski showed that there were infinitely many such
numbers k (all solutions to a family of congruences),
but he did not explicitly give a numerical example.
The congruences provided a sufficient, but not
necessary, condition for an integer to be a Sierpinski
number. Of course Sierpinski also asked what the
smallest such number might be--determining this
number is called the Sierpinski problem.

If the congruences proposed by Sierpinski are
solved, a 19-digit number k is obtained as
their smallest solution. The much smaller example
k = 78557, now conjectured to be the smallest
Sierpinski number, was found by John Selfridge in 1962. This is a Sierpinski number because it
has the "covering set:"

{3, 5, 7, 13, 19, 37, 73}

Why call this a covering set?
Because every number
of the form 78557.2n+1
is divisible by one of these small primes.
So these seven primes cover every possible
case!

n

covering set

78557

{3, 5, 7, 13, 19, 37, 73}

271129

{3, 5, 7, 13, 17, 241}

271577

{3, 5, 7, 13, 17, 241}

322523

{3, 5, 7, 13, 37, 73, 109}

327739

{3, 5, 7, 13, 17, 97, 257}

482719

{3, 5, 7, 13, 17, 241}

575041

{3, 5, 7, 13, 17, 241}

603713

{3, 5, 7, 13, 17, 241}

903983

{3, 5, 7, 13, 17, 241}

934909

{3, 5, 7, 13, 19, 73, 109}

965431

{3, 5, 7, 13, 17, 241}

1259779

{3, 5, 7, 13, 19, 73, 109}

1290677

{3, 5, 7, 13, 19, 37, 109}

1518781

{3, 5, 7, 13, 17, 241}

1624097

{3, 5, 7, 13, 17, 241}

1639459

{3, 5, 7, 13, 17, 241}

1777613

{3, 5, 7, 13, 17, 19, 109, 433}

2131043

{3, 5, 7, 13, 17, 241}

It is conjectured that 78557 is the smallest Sierpinski
number because for most of the smaller numbers
we can easily find a prime (in fact, for about 2/3rds
of the numbers k there is a prime with
n less than 9). The rest were then
checked for small covering sets and none were found,
so we expect to find a prime for the remaining
forms.

To prove the Sierpinski conjecture, "all" you need
to do is: for each of the following values of
k, find an exponent n
which makes k.2n+1
prime (for that particular value of k):

10223, 21181, 22699, 24737, 55459, and 67607.

Showing whether the list above is complete or
not will take much more of this same type of
prime finding.

The most recent and spectacular
success on the Sierpinski problem is by the
"Seventeen or Bust" project which has recently
found primes for the multipliers 4847, 5359, 19294, 27653, 28433, 33661, 44131, 46157,
54767, 65567, and 69109! The prime they found for 19249 has 3,918,990 digits. It is expected that the primes for the last few multipliers will be very very large.

In 1956 Riesel studied the corresponding
problem for numbers of the form
k.2n-1
(see Riesel numbers).